Through many cultures, star polygons were used as sacred
symbols with the star of David and the Sri Yantra Hindu patterns shown
in Figure 1a and 1b as two examples.
The fact that Venus traverses a five-pointed star over an eight-year cycle
in the heavens as seen from the Earth, shown in Figure
1c, was known to ancient civilizations. Also the designs of ancient
sacred geometry use a small vocabulary of proportions such as Ö
2, Ö 3, the golden mean t
= (1+Ö 5)/2 and the silver mean
q
= 1 + Ö 2. I will show that all of these
constants can be related to the edge lengths of star polygons and that
they are ultimately related to a sequence of numbers called silver means
the first of which is the golden mean. These silver means will also be
shown to be generalizations of the imaginary number i in some sense.
The geometry of the star heptagon will be found to be particularly interesting.
I have previously shown that star polygons are also related to the chaotic
dynamics of the logistic equation [Kappraff 2002].

A regular star polygon is denoted by the symbol {n/k}
where n is the number of vertices and edges while k indicates
that each vertex is connected to the k-th vertex from it in a clockwise
direction. Under certain conditions a regular star polygon can be drawn
in a single stroke such as polygon {5/2} in Figure 2a
or in more than one stroke as polygon {6/2} in Figure 2b.
Is there a simple rule that predicts whether the polygon is irreducible
(can be drawn in a single stroke) or reducible (cannot be drawn in a single
stroke)? The answer is simple: a polygon {n/k} will be connected
if and only if n and k
are relatively prime.

Figure 2The {5/2} star can be drawn in a single stroke; the {6/2} star cannot be drawn in a single stroke.

Only connected stars are considered to be star polygons.
For example, {6/2} actually corresponds to the connected 3-gon {3/1}. As
a result of the fact that n and k must be relatively prime,
there are as many regular n-gons as there are positive integers
relatively prime to n denoted by the Euler phi-function f
(n). For example, there are f (7) = 6
species of star 7-gons and f (12) = 4 species
of 12-gons. The three clockwise oriented 7-gons are shown in Figure
3 with three identical but counterclockwise oriented 7-gons not shown.
Of the 12-gons illustrated in Figure 4 only {12/1}
and {12/5} are actual 12-gons. They are the two clockwise oriented 12-gons
with the two counterclockwise 12-gons not shown.

The edges of a star n-gon are the diagonals
of the regular n-gon. By determining the lengths of the n–3
diagonals that can be drawn from a given vertex of a regular n-gon,
including the edge, we are also determining the lengths of the edges of
various species of star n-gons. Furthermore, when all polygons are
normalized to radius equal to one, the diagonals and edges of any m-gon
reappear in n-gons whenever n is a multiple of m.
For example, in Figure 4 the edges of the triangle,
square and hexagon appear as diagonals of a regular 12-gon. In this way
we are able to determine all of the star m-gons related to an n-gon
by considering all of the factors of n. For example, consider the
12-gon. Its factor tree is,

3 ® 6 ®
12 ¬ 4 ¬
2.
(1)

The edges of the triangle reappear as diagonals of the
hexagon and the 12-gon. Also the edges of the square reappear within the
12-gon. The factor 2 corresponds to the diameter {12/6} in Figure
4 and can be thought of as a polygon with two edges, referred to as
a digon {2/1}. In some limiting sense a single vertex can be thought
to be a polygon with a single edge of zero length {1/1}. If we add up the
total number of star polygons corresponding to these factors we get

f (1) + f
(2) + f (3) + f (4)
+ f (6) + f (12)
= 12.
(2)

These are all the star polygons related to the 12-gon.
In addition to the single vertex and the digon, five of these have their
edges oriented clockwise while five mirror images have retrograde edges.
It can be shown that, in general,

where the summation is taken over the integers that divide
evenly into n.

Star polygons are connected to number theory in many ways.
One striking example is due to H.S.M. Coxeter in which the star polygon
gives an elegant proof of Wilson’s Theorem.

Proof: For any p-gon there are clearly (p–1)!
distinct p-gons. Since f (p) =
p–1,
p–1
of them are regular while the others are irregular. Since the regular polygons
are invariant under rotation while the irregular ones are not, each irregular
configuration gives a new polygon under rotation about the center through
the angle between adjacent vertices. So if there are N classes of
irregular polygons, when these are rotated about the p vertices,
there are Np irregular polygons. Therefore,

Starting from the left in Table 1 the
diagonals: 111…, 123…,136…, etc. appear as columns in which each
successive column is displaced, in a downward direction, from the previous
column by two rows. The rows of Table 2 can be seen to
be the diagonals of Pascal’s triangle related to the Fibonacci sequence
with the sum of the elements of row n being the n-th Fibonacci
number. This table, referred to as the Fibonacci-Pascal Triangle or FPT,
is associated with the coefficients of a series of polynomials, F1(n),
(the superscripts are the exponents of the polynomials) [Adamson 1994].

n/k

1

2

3

4

5

Sum

0

10

1

1

1

11

1

x

2

12

10

2

x2 + 1

3

13

21

3

x3 + 2x

4

14

32

10

5

x4 + 3x2 + 1

5

15

43

31

8

x5 + 4x3 + 3x

6

16

54

62

10

13

x6 + 5x4 + 6x2
+ 1

7

17

65

103

41

21

x7+6x5+10x3+4x1

8

18

76

154

102

10

34

x8 + 7x6 + 15x4
+ 10x2 + 1

etc.

Table 2.Fibonacci-Pascal
TriangleEach column in Table 2 begins with a 1.
The numbers (not the exponents) are generated by the recursion relation:

(n,k) = (n-1,k) + (n-2,k-1)

where (n,k) denotes the number in the n-th
row and k-th column. For example, (7,3) = (6,3) + (5,2) or 10 =
6 + 4. Also, beginning with 1 and x, each Fibonacci polynomial F1(n)
is gotten by multiplying the previous one, F1(n-1)
by x and adding it to the one before it F1(n–2),
e.g., F1(3) = xF1(2) + F1(1)
or x3 + 2x = x(x2 + 1)
+ x.

Letting x = 1 in the polynomials yields the Fibonacci
sequence: 1 1 2 3 5 8 ... The ratios of successive numbers in this series
converge to the solution to x – 1/x = 1 or the golden mean
t
which I shall also refer to as the first silver mean of type 1 or
SM1(1).

Letting x = 2 yields Pell's sequence: 1 2 5 12
29 70 ... (e.g., to get a number from this sequence, double the preceding
term and add the one before it). The ratio of successive terms converges
to the solution of x – 1/x = 2 or the number
q
= 1 + Ö 2 = 2.414213..., referred to as
the silver mean or more specifically as the 2nd silver mean of type
1,
SM1(2).

Letting x = 3 yields the sequence: 1 3 10 33 109
... (e.g., to get a number from the sequence, triple the proceeding term
and add the one before it). The ratio of successive terms converges to
the solution to x – 1/x = 3 which is the 3rd Silver Mean
of type 1 or SM1(3).

In general letting x = N, where N
is either a positive or negative integer, leads to an approximate geometric
sequence for which,

xk+1 = Nxk
+ xk-1 ,

and whose ratio of successive terms is SM1(N)
which satisfies the equation,

x – 1/x = N.
(3a)

When the polynomials in Table 2 have
alternating signs they are denoted by F2(x), and
upon letting x = N,

xk+1 = Nxk
– xk-1 ,

and whose ratio of successive terms is SM2(N),
silver means of the second kind, which satisfies the equation,

x + 1/x = N.
(3b)

4. Lucas Version of Pascal’s Triangle

Fibonacci and Lucas sequences are intimately connected.
The standard Fibonacci sequence, {Fn}, is: 1 1 2 3 5
8 13 … while the standard Lucas sequence, {Ln}, is 1
3 4 7 11 18... where Ln = Fn-1 + Fn+1.
Adamson has discovered another variant of Pascal’s triangle related to
the Lucas sequence. In fact this Lucas-Pascal Triangle or LPT demonstrates
that silver mean constants and sequences are part of an interrelated whole.
Along with the FPT these tables are carriers of all of the significant
properties of the silver means.

To construct the LPT, create a new "Pascal’s triangle"
with 1’s along one edge and 2’s along the other as shown in Table
3.

2

2

1

2

3

1

2

5

4

1

2

7

9

5

1

...

Table 3.A Generalized
Pascal’s Triangle

As before each diagonal becomes a column of the LPT
in which the elements in each successive column are displaced downwards
by two rows. The exponents of the corresponding polynomials are sequenced
as for the FPT. You will notice that the numbers in each row sum
to the Lucas sequence and therefore I refer to the associated polynomials
as Lucas polynomials L1(n) to distinguish these
from Lucas polynomials with alternating signs denoted by L2(x).

n/k

1

2

3

4

Sum

0

20

2

2

1

11

1

x

2

12

20

3

x2 +2

3

13

31

4

x3+3x

4

14

42

20

7

x4+4x2+2

5

15

53

51

11

x5+5x3+5x

6

16

64

92

20

18

x6+6x4+9x2+2

7

17

75

143

71

29

x7+7x5+14x3+7x

etc.

Table 4.Lucas-Pascal’s
Triangle

Beginning with 2 and x, a Lucas polynomial is generated
by the recursive formula:

L1(n) = xL1(n-1)
+ L1(n-2),

for example L1(3) = xL(2) +L1(1)
or x3 + 3x = x(x2 + 2)
+ x .

Setting x = 1,2,3,… in the Lucas polynomials generates
a set of Generalized Lucas sequences that are related to the SM1(N)
constants. Many interesting properties of these sequences are described
in [Kappraff 2002]. They also lead directly to an
infinite set of generalized Mandelbrot sets [Kappraff
2002].

5. The relationship between Fibonacci and Lucas polynomials and regular star polygons.

We have discovered a simple relationship between the roots
of both the Fibonacci and Lucas polynomials with alternating signs and
the diagonals of regular polygons when the radii of the polygons are
taken to be 1 unit.

For odd n, the positive roots of the n-th
Lucas
polynomial, L2(n), with alternating signs
equal the distinct diagonal lengths dk for k>1
and edge d1 of the n-gon of radius 1 unit where

The results for several polygons are summarized in Table
5. The diagonals are normalized to an edge value of 1 unit by dividing
by d1.

Table 5. Lengths of
Normalized Diagonals of n-gons

We find the curious property that both the sum and product
of the squares of the diagonals of an n-gon (including the edge)
equals an integer and that this integer equals n for odd values
of n. For example, d12 + d22=
5 and d12d22 =
5 for the pentagon, while, d12 + d22
+ d32 = 7 and d12d22d32=
7 for the heptagon.

Not only are the diagonals of regular polygons determined
by Equations 4 and 5, but the areas
A of the regular n-gons with unit radii are computed from
the elegant formula,

From this equation the square is found to have area 2
units while the 12-gon has area 3 units. It can also be determined that
if n approaches infinity, then A approaches p
, the area of a unit circle.

Notice that the key numbers in the systems of proportions
based on various polygons present themselves in Table 5: t–pentagonal
system; q and Ö
2 –octagonal; Ö 3, 1+ Ö
3, and 2+Ö 3 –dodecahedral; r
and s –heptagonal, and these are pictured in
Figure 6.

Since the unique diagonals of an n-gon correspond
to the roots of a polynomial, the fact that these diagonals recur in any
m-gon
where n is a multiple of m serves to factor the polynomial
into polynomials of smaller degree with integer coefficients. For example,
the polynomial F2(9) of the 10-gon factors into the product
of L2(5) of the 5-gon and F2(4), i.e.,
F2(9)
= L2(5)×F2(4) or,

x9 – 8x7 + 21x5
–
20x3 + 5x = (x4 – 3x2
+
1)(x5 – 5x3 + 5x).

We can state this result as a theorem:

Theorem 1: The polynomial F2(2n–1)
of any 2n-gon factors into the product of L2(n)
and F2(n–1).

By the same reasoning as for the 10-gon, the factor tree
of Expression 1 can be used to factor F2(11), the polynomial
representing the 12-gon. Of the six unique diagonals of the 12-gon, one
occurs in the 3-gon (equilateral triangle), an additional one appears in
the 6-gon (hexagon), another appears in the 4-gon (square), and two additional
diagonals occur in the 12-gon. By Theorem 1, the polynomial
of the 12-gon factors into,

F2(11) = L2(6) ×
F2(5)

corresponding to the factoring by the hexagon polynomial
F2(5).
The hexagon polynomial can then be decomposed further as,

F2(5) = L2(3) ×
F2(2)

corresponding to factoring by the triangle L2(3).
These two factorizations can be combined to obtain,

The diagonal (edge) of the triangle comes from L2(3),
the additional diagonal (edge) of the hexagon from F2(2),
the diagonal of the square is the root of the first factor of L2(6)
while the two additional diagonals of the 12-gon are the roots of the second
factor of L2(6). Finally, the diagonal of the digon is
the diameter of the 12-gon. This accounts for the six distinct diagonals
of the 12-gon.

In what follows the symbol Dk will be
used for diagonals of regular polygons normalized to a unit edge
rather

7. Additive properties of the diagonal lengths

Similar to t and q
, the diagonals of each of these systems of n-gons have additive
properties. Steinbach has derived the following Diagonal Product Formula
(DPF) that defines multiplication of the diagonal lengths in terms
of their addition [Steinbach 1997, 2000],

DhDk
= , where h£k.

where the diagonals have been normalized to polygons with
edges of D1 = 1 unit. It is helpful to write these identities
in an array as follows:

D22 = 1 + D3

D32 = 1 + D3
+ D5

D42 = 1 + D3
+ D5 + D7

(6)

D2D3 = D2
+ D4

D3D4 = D2
+ D4 + D6

D4D5 = D2
+ D4 + D6 +
D8

D2D4 = D3
+ D5

D3D5 = D3
+ D5 +D7

...

D2D5 = D4
+ D6

...

These formulas are applied to the pentagon and the heptagon.

Example 1:

For the pentagon, D2 = D3
= t and these relationships reduce to the single
equation,

D22 = 1 + D3
or
t2
= 1 + t

Example 2:

The proportional system based on the heptagon is particularly
interesting [Steinbach 1997], [Ogawa
1990]. For the heptagon, D2=D5=r
and D3=D4=s
and these relationships reduce to the four equations,

D22 = 1+ D3

or

r2 = 1 + s

D2D3 = D2
+ D4

or

rs = r
+ s

D32 = 1 + D3
+ D5

or

s2 = 1 + s
+ r

What is astounding is that not only are the products of
the diagonals expressible as sums but so are the quotients. Table 6 illustrates
the quotient table for the heptagon.

1

r

s

1

1

1 + r – s

s – r

r

r

1

r – 1

s

s

s – 1

1

Table 6Ratio of diagonals
(left /top)

As a result of DPF and the quotient laws, Steinbach
has discovered that the edge lengths of each polygon form an algebraic
system closed under the operations of addition, subtraction, multiplication,
and division. Such algebraic systems are known as fields and he
refers to them as golden fields.

8. The Heptagonal System

The heptagonal system is particularly rich in algebraic
and geometric relationships. The additive properties of DPF and
Table 6 for the heptagon are summarized:

r+ s
= rs
(Compare this with t +t2
= tt2)

1/r + 1/s
= 1 (Compare this with 1/t + 1/t2
= 1)

r2 = 1 + s

s2 = 1 + r
+ s

r /s
= r– 1

s /r
= s– 1
(7)

1/s = s–
r

1/r = 1 + r
– s

The algebraic properties of each system of proportions
are manifested within the segments of the n-pointed star (the n-polygon
with all of its diagonals) corresponding to that system. For example, the
pentagonal system of proportions is determined by the 5-star while the
octagonal system is determined by the 8-star. Figure 7
illustrates the family of star heptagons. Notice that the short diagonal
of length r (the edge is 1 unit) and the long
diagonal of length s are subdivided into the
following segments depending on r and s
:

r = 1/r
+ 1/rs + 1/s2
+ 1/rs + 1/r and,

s = 1/s
+ r /s2
+ 1/s2 + 1/rs
+ 1/s2 + r
/s2 + 1/s

Figure 7A heptagon with all of its diagonals. Its
principal diagonals r and s
are shown to be rationally subdivided.

Thus we see at the level of geometry that the graphic
designer encounters the same rich set of relationships as does the mathematician
at the level of symbols and algebra.

The following pair of intertwining geometric s
-sequences and corresponding Fibonacci-like integer series exhibit these
additive properties:

...

1/s

r /s

1

r

s

sr

s2

s2r

s3

s3r

s4

...

(8a)

1

1

1

2

3

5

6

11

14

25

31

(8b)

The integer series is generated as follows:

Determine the first five terms xyzuv beginning with
111

Let y+z=u and u+x=v,
i.e., 1+1=2 and 2+1=3 to obtain 11123

Repeat step 2 beginning with the zuv, i.e., from 123,
2+3=5 and 5+1=6 to obtain 12356

Continue

The ratio of successive terms of this sequence equals, alternatively
r
and s /r while the
ratios of successive terms of the integer series asymptotically approaches
r
and s /r , e.g.,
25/14 = 1.785… »r while 31/25 = 1.24 »s
/r . Also s is obtained
as the product of these ratios, i.e., 31/14 = 2.214 »s
. Just as every power of the golden mean t can
be written as a linear combination of 1 and t
with the Fibonacci numbers as coefficients [Kappraff
2001], every power of s can be written as
the following linear combinations of 1, r ,
s where the integers of Sequence
8b appear as the coefficients:

s = 1s
+ 0r + 0

s2 = 1s
+ 1r + 1

s3 = 3s
+ 2r + 1
(9)

s4 = 6s
+ 5r + 3

s5 = 14s
+ 11r + 6

s6 = 31s
+ 25r + 14

…

Notice that the first coefficient in the equation for
sn+1
is the sum of the three coefficients of the equation for sn
while the second coefficient is the sum of the first two coefficients and
the last coefficient is the same as the first of the previous equation,
e.g., in the equation for s4: 6=3+2+1,
5=3+2, and 3=3.

A geometric analogy to the golden mean can be seen by
considering the pair of rectangles of proportions r:
1 and s : 1 in Figure 8. By
removing a square from each, we are left in both cases with rectangles
of proportion r:
s
although oriented differently.

(a)

(b)

Figure 8When a square is removed from a 1: r
and a 1: s rectangle the leftover portion is a r
: s rectangle in different orientations.

9. Self-referential Properties of the Silver Mean Constants

In general the equation T(x) = x
expresses a fixed point or what I refer to as a self-referential
relationship. Replacing x by T(x) gives T(T(x))
= x or TTx = x. Continuing this process results in
,

Although this infinite compound fraction has no mathematical
meaning, the infinite process can be defined to be the imaginary numbers
±i
since
these are the solutions to –1/x = x.

We now come to a set of self-referential statements related
to the SM1 and SM2 constants. These
constants are solutions to the self-referential equations T(x) =
x
where,

T(x) = N + 1/x and T(x)
= N - 1/x.

If N = 0 in the second of these transformations,
TTT...
is identified with the imaginary number i. So in a sense, the silver
means are generalizations of i. The solutions
x can be shown
to be the two infinite processes,

TTT... = N+(1/(1+N/(1+N)))...
and TTT... = N-(1/(1-N/(1-N)))...

These are continued fraction representations of the silver
mean constants of types 1 and 2. SM1(N) = [N;`N]
in continued fraction notation. SM2(N) = [N;`N]-
are expressed in terms of another form of continued fraction not discussed
in this book.

10. Conclusion

With the aid of Pascal’s triangle, the golden mean and
Fibonacci sequences were generalized to a family of silver means. The Lucas
sequence was then generalized with the aid of a close variant of the Pascal’s
triangle. These generalized golden means and generalized F- and
L-sequences
were shown to form a tightly knit family with many properties of number.
Perhaps it is for this reason that they occur in many dynamical systems.
The numerical properties of the silver mean constants are the result of
their self-referential properties which, in turn, derive from their relationship
to the imaginary number i. We have shown that all systems of proportion
are related to a set of polynomials derived from Pascal’s triangle. These
systems are related to both the edges of various species of regular star
polygon and the diagonals of regular n-gons, and they share many
of the additive properties of the golden mean. The heptagon was illustrated
in detail.